1. Fall 2003Costas Busch - RPI1 Linear Grammars Grammars with at most one variable at the right side of a production Examples:

2. Fall 2003Costas Busch - RPI2 A Non-Linear Grammar Grammar : Number of in string

3. Fall 2003Costas Busch - RPI3 Another Linear Grammar Grammar :

4. Fall 2003Costas Busch - RPI4 Right-Linear Grammars All productions have form: Example: or string of terminals

5. Fall 2003Costas Busch - RPI5 Left-Linear Grammars All productions have form: Example: or string of terminals

6. Fall 2003Costas Busch - RPI6 Regular Grammars

7. Fall 2003Costas Busch - RPI7 Regular Grammars A regular grammar is any right-linear or left-linear grammar Examples:

8. Fall 2003Costas Busch - RPI8 Observation Regular grammars generate regular languages Examples:

9. Fall 2003Costas Busch - RPI9 Regular Grammars Generate Regular Languages

10. Fall 2003Costas Busch - RPI10 Theorem Languages Generated by Regular Grammars Regular Languages

11. Fall 2003Costas Busch - RPI11 Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language

12. Fall 2003Costas Busch - RPI12 Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar

13. Fall 2003Costas Busch - RPI13 Proof – Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular

14. Fall 2003Costas Busch - RPI14 The case of Right-Linear Grammars Let be a right-linear grammar We will prove: is regular Proof idea: We will construct NFA with

15. Fall 2003Costas Busch - RPI15 Grammar is right-linear Example:

16. Fall 2003Costas Busch - RPI16 Construct NFA such that every state is a grammar variable: special final state

17. Fall 2003Costas Busch - RPI17 Add edges for each production:

18. Fall 2003Costas Busch - RPI18

19. Fall 2003Costas Busch - RPI19

20. Fall 2003Costas Busch - RPI20

21. Fall 2003Costas Busch - RPI21

22. Fall 2003Costas Busch - RPI22

23. Fall 2003Costas Busch - RPI23 Grammar NFA

24. Fall 2003Costas Busch - RPI24 In General A right-linear grammar has variables: and productions: or

25. Fall 2003Costas Busch - RPI25 We construct the NFA such that: each variable corresponds to a node: special final state

26. Fall 2003Costas Busch - RPI26 For each production: we add transitions and intermediate nodes ………

27. Fall 2003Costas Busch - RPI27 For each production: we add transitions and intermediate nodes ………

28. Fall 2003Costas Busch - RPI28 Resulting NFA looks like this: It holds that:

29. Fall 2003Costas Busch - RPI29 The case of Left-Linear Grammars Let be a left-linear grammar We will prove: is regular Proof idea: We will construct a right-linear grammar with

30. Fall 2003Costas Busch - RPI30 Since is left-linear grammar the productions look like:

31. Fall 2003Costas Busch - RPI31 Construct right-linear grammar Left linear Right linear

32. Fall 2003Costas Busch - RPI32 Construct right-linear grammar Left linear Right linear

33. Fall 2003Costas Busch - RPI33 It is easy to see that: Since is right-linear, we have: Regular Language Regular Language Regular Language

34. Fall 2003Costas Busch - RPI34 Proof - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by some regular grammar

35. Fall 2003Costas Busch - RPI35 Proof idea: Let be the NFA with. Construct from a regular grammar such that Any regular language is generated by some regular grammar

36. Fall 2003Costas Busch - RPI36 Since is regular there is an NFA such that Example:

37. Fall 2003Costas Busch - RPI37 Convert to a right-linear grammar

38. Fall 2003Costas Busch - RPI38

39. Fall 2003Costas Busch - RPI39

40. Fall 2003Costas Busch - RPI40

41. Fall 2003Costas Busch - RPI41 In General For any transition: Add production: variableterminalvariable

42. Fall 2003Costas Busch - RPI42 For any final state: Add production:

43. Fall 2003Costas Busch - RPI43 Since is right-linear grammar is also a regular grammar with